De Morgan's Laws (Set Theory)/Relative Complement

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Theorem

Let $S, T_1, T_2$ be sets such that $T_1, T_2$ are both subsets of $S$.


Then, using the notation of the relative complement:

Complement of Intersection

$\relcomp S {T_1 \cap T_2} = \relcomp S {T_1} \cup \relcomp S {T_2}$


Complement of Union

$\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$


General Case

Let $S$ be a set.

Let $T$ be a subset of $S$.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.


Then:

Complement of Intersection

$\displaystyle \complement_S \left({\bigcap \mathbb T}\right) = \bigcup_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$


Complement of Union

$\displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$


Family of Sets

Let $S$ be a set.

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.


Then:

Complement of Intersection

$\displaystyle \relcomp S {\bigcap_{i \mathop \in I} \mathbb S_i} = \bigcup_{i \mathop \in I} \relcomp S {S_i}$


Complement of Union

$\displaystyle \relcomp S {\bigcup_{i \mathop \in I} \mathbb S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$


Source of Name

This entry was named for Augustus De Morgan.